Numerical Algorithms最新文献

In this paper, we study the numerical solutions of the generalized inverse eigenvalue problem (for short, GIEP). Motivated by Ulm’s method for solving general nonlinear equations and the algorithm of Aishima (J. Comput. Appl. Math. 367, 112485 2020) for the GIEP, we propose here an Ulm-like algorithm for the GIEP. Compared with other existing methods for the GIEP, the proposed algorithm avoids solving the (approximate) Jacobian equations and so it seems more stable. Assuming that the relative generalized Jacobian matrices at a solution are nonsingular, we prove the quadratic convergence property of the proposed algorithm. Incidentally, we extend the work of Luo et al. (J. Nonlinear Convex Anal. 24, 2309–2328 2023) for the inverse eigenvalue problem (for short, IEP) to the GIEP. Some numerical examples are provided and comparisons with other algorithms are made.

In this article, we deal with nonconvex fractional programming problems involving E-differentiable functions ((FP_E)). The so-called E-Karush-Kuhn-Tucker sufficient E-optimality conditions are established for nonsmooth optimization problems under E-univexity hypothesis. The established optimality conditions are explained with a numerical example. The so-called vector dual problem in the sense of Schaible ((SD_E)) involves E-differentiable functions for ((FP_E)) is defined under E-univexity hypothesis.

In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great precision, even for large ill-conditioned matrices. In this framework, the present work provides the factorization of the collocation matrices of Newton bases—of relevance when considering the Lagrange interpolation problem—together with an algorithm that allows to numerically compute it to high relative accuracy. This further allows to determine the coefficients of the interpolating polynomial and to compute the singular values and the inverse of the collocation matrix. Conditions that guarantee high relative accuracy for these methods and, in the former case, for the classical recursion formula of divided differences, are determined. Numerical errors due to imprecise computer arithmetic or perturbed input data in the computation of the factorization are analyzed. Finally, numerical experiments illustrate the accuracy and effectiveness of the proposed methods with several algebraic problems, in stark contrast with traditional approaches.

In this paper, we intend to formulate and solve Cauchy problems for the Brinkman equations governing the flow of fluids in porous media, which have never been investigated before in such an inverse formulation. The physical scenario corresponds to situations where part of the boundary of the fluid domain is hostile or inaccessible, whilst on the remaining friendly part of the boundary we prescribe or measure both the fluid velocity and traction. The resulting mathematical formulation leads to a linear but ill-posed problem. A convergent algorithm based on solving two sub-sequences of mixed direct problems is developed. The direct solver is based on the method of fundamental solutions which is a meshless boundary collocation method. Since the investigated problem is ill-posed, the iterative process is stopped according to the discrepancy principle at a threshold given by the amount of noise with which the input measured data is contaminated in order to prevent the manifestation of instability. Results inverting both exact and noisy data for two- and three-dimensional problems demonstrate the convergence and stability of the proposed numerical algorithm.

Nonlinear coupled reaction-diffusion systems often arise in cooperative processes of chemical kinetics and biochemical reactions. Owing to these potential applications, this article presents an efficient and simple meshless approximation scheme to analyze the solution behavior of a two-dimensional coupled Brusselator system. On considering radial basis functions in the localized settings, meshless shape functions owing Kronecker delta function property are constructed to discretize the spatial derivatives in the time-dependent partial differential equation (PDE). A system of first-order ordinary differential equations (ODEs), obtained after spatial discretization, is then integrated in time via a high-order ODE solver. The proposed scheme’s convergence, stability, and efficiency are theoretically established and numerically verified on several benchmark problems. The outcomes verify reliability, accuracy, and simplicity of the proposed scheme against the available methods in the literature. Some recommendations are made regarding time-step size under different node distributions and RBFs.

Integrodifference equations are discrete-time counterparts to reaction-diffusion equations and have various applications in, e.g., theoretical ecology. Their behavior is often illustrated using numerical simulations, which require a spatial discretization. In this paper, we establish that periodic solutions to time-periodic integrodifference equations, their stability and their Floquet spectrum persist under discretization of Nyström-type, which replaces integrals by quadrature or cubature rules. Moreover, it is shown that the convergence rates of the particular integration rules are preserved. By means of a typical model from theoretical ecology, these results are demonstrated in terms of a numerical continuation for periodic solutions and their Floquet multipliers.

We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.

For Hamiltonian systems with non-canonical structure matrices, a new family of fourth-order energy-preserving integrators is presented. The integrators take a form of a combination of Runge–Kutta methods and continuous-stage Runge–Kutta methods and feature a set of free parameters that offer greater flexibility and efficiency. Specifically, we demonstrate that by carefully choosing these free parameters, a simplified Newton iteration applied to the integrators of order four can be parallelizable. This results in faster and more efficient integrators compared with existing fourth-order energy-preserving integrators.

We implement an Augmented Lagrangian method to minimize a constrained least-squares cost function designed to find sparse polyadic decompositions with elements of bounded maximal value of matrix multiplication tensors. We use this method to obtain new decompositions and parameter families of decompositions. Using these parametrizations, faster and more stable matrix multiplication algorithms are discovered.

Radial basis function finite difference (RBF-FD) discretization has recently emerged as an alternative to classical finite difference or finite element discretization of (systems) of partial differential equations. In this paper, we focus on the construction of preconditioners for the iterative solution of the resulting linear systems of equations. In RBF-FD, a higher discretization accuracy may be obtained by increasing the stencil size. This, however, leads to a less sparse and often also worse conditioned stiffness matrix which are both challenges for subsequent iterative solvers. We propose to construct preconditioners based on stiffness matrices resulting from RBF-FD discretization with smaller stencil sizes compared to the one for the actual system to be solved. In our numerical results, we focus on RBF-FD discretizations based on polyharmonic splines (PHS) with polynomial augmentation. We illustrate the performance of smaller stencil preconditioners in the solution of the three-dimensional convection-diffusion equation.